Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $a \neq 0$. $y = \dfrac{a + 1}{-3a - 12} \times \dfrac{a^2 + 13a + 36}{-2a - 2} $
Solution: First factor the quadratic. $y = \dfrac{a + 1}{-3a - 12} \times \dfrac{(a + 4)(a + 9)}{-2a - 2} $ Then factor out any other terms. $y = \dfrac{a + 1}{-3(a + 4)} \times \dfrac{(a + 4)(a + 9)}{-2(a + 1)} $ Then multiply the two numerators and multiply the two denominators. $y = \dfrac{ (a + 1) \times (a + 4)(a + 9) } { -3(a + 4) \times -2(a + 1) } $ $y = \dfrac{ (a + 1)(a + 4)(a + 9)}{ 6(a + 4)(a + 1)} $ Notice that $(a + 1)$ and $(a + 4)$ appear in both the numerator and denominator so we can cancel them. $y = \dfrac{ \cancel{(a + 1)}(a + 4)(a + 9)}{ 6\cancel{(a + 4)}(a + 1)} $ We are dividing by $a + 4$ , so $a + 4 \neq 0$ Therefore, $a \neq -4$ $y = \dfrac{ \cancel{(a + 1)}\cancel{(a + 4)}(a + 9)}{ 6\cancel{(a + 4)}\cancel{(a + 1)}} $ We are dividing by $a + 1$ , so $a + 1 \neq 0$ Therefore, $a \neq -1$ $y = \dfrac{a + 9}{6} ; \space a \neq -4 ; \space a \neq -1 $